Simple systems with anomalous dissipation and energy cascade

Jonathan C. Mattingly, Toufic Suidan, Eric Vanden Eijnden

Research output: Contribution to journalArticle

Abstract

We analyze a class of dynamical systems of the type ȧn(t) = cn-1 an-1(t) - cn an+1(t) + f n(t), n ∈ ℕ, a 0=0, where f n (t) is a forcing term with fn(t) ≠ = 0 only for ≤n n* < ∞ and the coupling coefficients c n satisfy a condition ensuring the formal conservation of energy 1/2 Σn |a n(t)|2. Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely E |an|2 scales as n as n→∞. Here the exponents α depend on the coupling coefficients c n and E denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.

Original languageEnglish (US)
Pages (from-to)189-220
Number of pages32
JournalCommunications in Mathematical Physics
Volume276
Issue number1
DOIs
StatePublished - Nov 2007

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Cascade
Anomalous
Dissipation
cascades
dissipation
Forcing Term
coupling coefficients
Turbulence
Coefficient
Dynamical system
Energy
dynamical systems
Inviscid Limit
Equilibrium Measure
Passive Scalar
turbulence
Energy Spectrum
Burgers Equation
Burger equation
energy

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Simple systems with anomalous dissipation and energy cascade. / Mattingly, Jonathan C.; Suidan, Toufic; Vanden Eijnden, Eric.

In: Communications in Mathematical Physics, Vol. 276, No. 1, 11.2007, p. 189-220.

Research output: Contribution to journalArticle

Mattingly, Jonathan C. ; Suidan, Toufic ; Vanden Eijnden, Eric. / Simple systems with anomalous dissipation and energy cascade. In: Communications in Mathematical Physics. 2007 ; Vol. 276, No. 1. pp. 189-220.
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